Ab initio Electronic Structure

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Pasteur, Strasbourg France http://www-ipcms.

1 Ab initio Electronic Structure M. Alouani IPCMS, UMR 7504, Université Louis Pasteur, Strasbourg France In coll. with: B. Arnaud, O. Bengone, Y. Dappe, and S. Lebègue

2 1965 W. Kohn and L.J. Sham, Selfconsistent Equations including Exchange and Correlation Effects, Phys. Rev. 140, A (1965). ({ }) = ( ) ( ) E R Min E n i L. Hedin, New Method for Calculating the One-Particle Green s Function with Application to the Electron-Gas Problem, Phys. Rev 139, A (1965). n r Σ = igw J.W. Cooley and J. W. Tukey, Algorithm of the FFT, Math. Comput. 19, (1965). N 2 N log N 2

3 Goal: Describe properties of matter from ab initio methods. structural electronic vibrational PROPERTIES of materials optical magnetic

4 Main approximations: Born-Oppenhaimer Decouple the movement of the electrons and the nuclei. Hartree-Fock Approximation Treatment of the electron electron interactions. Density Functional Theory Treatment of the electron electron interactions. Pseudopotentials or all electron potentials Basis set Treatment of the (nuclei + core) valence. To expand the eigenstates of the hamiltonian. Numerical evaluation of matrix elements Supercells Efficient and self-consistent computations of H and S. Makes the calculation of materials possible (Bloch theorem)

5 Born-Oppenheimer Approximation Adiabatic or Born-Oppenheimer approximation decouple the electronic and nuclear degrees of freedom M p >> M e Nuclei much slower than the electrons v e >> v p Electrons in their ground state for any instantaneous ionic configuration. Solve electronic equations assuming fixed positions for nuclei (1) Move the nuclei as classical particles in the potential generated by the e - (2)

Density Functional Theory The Kohn-Sham ansatz replaces the many-body problem with an independent-particle problem All the properties of the system are completely determined given only the ground

6 Density Functional Theory The Kohn-Sham ansatz replaces the many-body problem with an independent-particle problem All the properties of the system are completely determined given only the ground state density n But no prescription to solve the difficult interacting many-body hamiltonian Ground state density of the many-body interacting system = Density of an auxiliary non-interacting independent particle system Kohn-Sham ansatz (never proven in general)

7 Density Functional Theory Density functional theory is the most widely used method today for electronic structure calculations because of the approach proposed by Kohn and Sham

8 Density Functional Theory One electron or independent particle model We assume that each electron moves independently in a potential created by the nuclei and the rest of the electrons. Actual calculations performed on the auxiliary independent-particle system

9 Density Functional Theory Like the derivation of the HF equations, we can use the variational theorem to find out the KS equations. This time we take a variation with respect to the electron density: { E n d 3 rn N } δ [ ] ε (r) = 0 * If we write (r) = (r) (r) and knowing that n ψ ψ δ E[ n] δ E[ n] = δn δn We then obtain from the variational theorem the KS SE: δ FHK [ n(r)] ( + Vext (r)) ψ( r) = εψ ( r) δn(r) If the exact form of the F HK is known then the problem is solved, since the SE will produce a selfconsistent ψ that determines the density, and hence the ground state total energy. The trick is to use the independent-particle kinetic energy which is given explicitly as a functional of the orbitals:

10 Density Functional Theory Then one has to rewrite the functional as The variational theorem gives then: The exchange-correlation energy has to obey certain rules: The rest: Exchangecorrelation 1. Like in the HFA, E XC can be written in terms of the exchange-correlation hole n xc : 2. Again, like in the HFA, the exchange-correlation hole has to fulfill the sum rule: This sum rule helps constrain the search for E XC.

The general form of E XC : Local density approximation (LDA): Density Functional Theory = 3 EXC[ n] n( r) ε XC[ n, r] d r є XC at r depends on the density shape n in the whole space.

11 The general form of E XC : Local density approximation (LDA): Density Functional Theory = 3 EXC[ n] n( r) ε XC[ n, r] d r є XC at r depends on the density shape n in the whole space. = 3 EXC[ n] n( r) ε XC ( n( r)) d r є XC at r depends only on the density at r (exact for a homogeneous electron gas). It is a function of the density and not a functional! Generalized Gradient Approximation (GGA) 3 EXC[ n] = n( r) ε XC ( n( r), n( r),...) d r є XC at r depends on the density and its gradient at r. The success of the LDA is certainly due to the fact that it satisfy the n xc sum rule, and being constructed from a homogeneous electron gas, it depends only on the spherical average of this hole density.

12 Density Functional Theory The Kohn-Sham equations must be solved selfconsistently The potential (input) depends on the density (output) I. Input: Structure, Atomic species II. Guess for input III. Compute the potential IV. Solve the Ks equations V. Compute the output density VI. If selfconsistent: output the tolal energy,forces, etc Yes Output quantities Energy, forces, stresses

13 Used approximations, basis-sets, potentials, etc, PAW

14 Density Functional Theory Accuracy of the XC functionals in the structural and electronic properties a B E c E gap LDA -1%, -3% +10, +40% +15% -50% GGA +1% -20%, +10% -5% -50% LDA: simplest approximation but accurate enough (structural properties, ). GGA: usually tends to overcompensate LDA results, but not always better.

15 Density Functional Theory In some cases, GGA is a must: DFT ground state of iron GGA GGA LSDA GGA LSDA P. Bagno et al., PRB 40, 1997 (1987) Results obtained with Wien2k. Courtesy of Karl H. Schwartz LSDA NM fcc GGA FM bcc Correct lattice constant Experiment FM bcc

16 GGA after LDA Picture provided by Ambrosch-Draxl

17 p Quasiparticle concept p Real particle Quasi particle p p Real horse Quasi horse R. D. Mattuck, a guide to Feynman Diagrams in the MB problem, Dover, 1976 A quasiparticle has an effective mass, selfenergy (energy and lifetime).

18 The PAW method The Projector augmented-wave (PAW) method was introduced by Blöchl, Ψ ( r ) = Ψ ( r ) + c Λ ϕ Λ ( r ) ϕ Λ ( r ) Λ

19 The PAW method continued... LDA potential and pseudopotential of Si in the (001) plan

20 The PAW method continued... Expectation value of a local one particle operator: O = Ψ O Ψ + n D ϕ O ϕ ϕ O ϕ n n i, j i j i j i, j Di, j Ψn pi p j n = Ψ n

21 Dielectric function of Si and GaAs The PAW (solid line) and the FPLMTO (dashed line) calculations are in good agreement. The local-field and the excitonic effects are not included in both calculations.

22 The DFT band-gap problem The band gap is defined as the energy difference between the Ionization energy and the electron affinity: ( N 1) ( N ) ( N ) ( N + 1) Eg = E E E E = I A where ( N ) E ±1 is the total energy the fondamental state of a system of electrons N ±1 Schlüter and Sham (PRL, 1983) showed that the DFT band gap is not equal to that of the quasiparticles ( N ) ( N ) ( N + 1) ( N ) = E ε = E E E = V V DFT g g g N + 1,DFT N,DFT XC XC

23 Static dielectric function of semiconductors The LDA overestimates the static dielectric function of semiconductors

24 LDA band gaps

25 Hartree-Fock approximation

26 Static dielectric function with local field effects The RPA static dielectric function with local field effects is much closer to the experimental results

27 The GWA, Hedin 1965

31 Removal of the double counting

33 Comparison with the plasmon-pole model

35 Direct integration versus Padé

36 Quasiparticle band structure of Si B. Arnaud et M.A., PRB 62, 3923 (00).

37 GW band gaps B. Arnaud et M.A., PRB 62, 3923 (00).

38 Metal-insulator transition in LiH Phase transition: NaCl CsCl LDA GGA GW_LDA GW_GGA S. Lebègue, M. A., B. Arnaud, W. E. Pickett, Europhys. Lett. 68, 846 (04);

39 Electronic structure of LiF B. Arnaud et M.A., PRB 62, 3923 (00).

40 Optical properties within the RPA The dielectric function within the RPA is given by: B. Arnaud et M.A., PRB 63, (01).

c k A = E A λ λ λ ck vk vck v c k λ vck v,c,k The macroscopic dielectric

41 Excitonic effects (Hanke and Sham, 1975) ( 3, 3, 4, 4 ) i δ ( 3, 4) δ ( 3, 4 ) W ( 3, 3 ) i δ ( 3, 3 ) δ ( 4, 4 ) V ( 3, 4) Ξ = ( qp qp ε ε ) ε ε A + vck Ξ v c k A = E A λ λ λ ck vk vck v c k λ vck v,c,k The macroscopic dielectric function: 2 2 4π 1 lim vk e iqr λ ε 2 ω = ck Avck δ ω E 2 Ω q ( ) ( ) q 0 λ v,c,k λ

42 Dielectric function of Silicon: B. Arnaud et M. A., PRB (01)

43 Ionic Insulators (KCl)

It consists of hexagonal graphite-like sheets but with an ABAB

44 Structure de h-bn h-bn graphite h-bn crystallizes in a hexagonal structure (D 6h4 group). It consists of hexagonal graphite-like sheets but with an ABAB stacking with boron atoms in layer A found directly below nitrogen atoms in layer B Nanotube of BN

45 Lattice constants and bulk modulus of h-bn Method a (a.u.) c (a.u.) B (Mbar) Present US-PP LCAO Expt ± Kern, Kresse, and Hafner, PRB B 59, 8551 (99) 2. Xu and Ching, PRB 44, 7787 (91) 3. Solozhenko, Will, and Elf, SSC 96, 1 (95) The underestimation of the c parameter and the bulk modulus is a result a flatter total energy at the equilibrium position. Bader analysis shows a charge transfer of 2.16 electrons from B to N.

46 Band gaps in ev (LDA versus GW compared to experiment) Method LDA Direct Indirect GW Direct Indirect Expt. Present PP a b c Expt. d e f ± a. Furthmüller, Hafner, Kresse, PRB 50, (94). b. Capellini et al. GaN and related mat. Edited by Dupuis et al, MRS, Pettsburg, 96. c. Capelline et al, PRB 64, (01). d. Hoffman, Doll, Eklund, PRB 30, 6051 (84) (optical reflectance). e. Zunger, Doll, Eklund, PRB 30, 5560 (76) (optical absorption). f. Watanabe, Taniguchi, Kanda, Nature Materials, 3, 404 (04) (optical absorption).

47 Band structure of h-bn within LDA ( ) and GW ( )

48 Optical spectra of h-bn within LDA and GW Expt: Mamy et al, J. Phys. Lett 42, 473 (81) Static dielectric function Without LF ε 4.76 ε 2.98 With LF B. Arnaud et al., PRL 96, (06).

49 Absorption spectrum of h-bn within LDA and GW Expt; Watanabe et al., Nature Mat. 3, 404 (2004). B. Arnaud et al., PRL 96, (06).

Two dimensional projections of the probability density state at 5.780 ev.

located slightly above a N atom Distribution of the electron relative to the hole in a

50 Two dimensional projections of the probability density state at ev. λ Ψ ( r,r ) h e 2 for the exciton Distribution of the electron relative to the hole located slightly above a N atom Distribution of the electron relative to the hole in a plane parallel to the c-axis passing through the line shown in the left panel. B. Arnaud et al., PRL 96, (06).

51 Conclusions and Perspectives Conclusions The PAW method is as powerful as any other allelectron method, but much faster The GW-PAW improves the band gaps of insulating materials The GW-PAW QP energies can be used to determine the optical spectra of materials Actinic effects are important for absorption spectra Perspectives GW for metals and magnetic materials Selfconsistent GW-DMFT (GdN work of Samir) Computing XAS and XMCD spectra Computing molecular systems (Transport see Bengone) Doped Nanotubes (Transport, Debbichi)

Density Functional Theory. Martin Lüders Daresbury Laboratory

Density Functional Theory. Martin Lüders Daresbury Laboratory Density Functional Theory Martin Lüders Daresbury Laboratory Ab initio Calculations Hamiltonian: (without external fields, non-relativistic) impossible to solve exactly!! Electrons Nuclei Electron-Nuclei